Georgia Institute of Technology College of Sciences

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov—He and Church—Ershov—Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also discuss some extensions of these ideas.  In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.

The complete flag variety is a fundamental object at the confluence of algebraic geometry, representation theory, and algebra. It is defined to be the space parametrizing certain chains of vector subspaces, and is intimately linked to Grassmannians, incidence varieties, and other important geometric objects of a representation-theoretic flavor. The problem of computing the cohomology of any line bundle on a flag variety in characteristic 0 was solved in the 1950's, culminating in the celebrated Borel--Weil--Bott theorem. The situation in positive characteristic is wildly different, and remains a wide-open problem despite many decades of study. After surveying this topic, I will speak about recent progress on a characteristic-free analogue of the Borel--Weil--Bott theorem through the lens of representation stability and the theory of polynomial functors. This "stabilization" of cohomology, combined with certain universal categorifications of the Jacobi-Trudi identity, has opened the door to concrete computational techniques whose applications include effective vanishing results for Koszul modules, yielding an algebraic counterpart for the failure of Green's conjecture for generic curves in arbitrary characteristic.

Krylov subspace methods (KSMs) are among the most widely used algorithms for a number of core linear algebra tasks. However, despite their ubiquity throughout the computational sciences, there are many open questions regarding the remarkable convergence of commonly used KSMs. Moreover, there is still potential for the development of new methods, particularly through the incorporation of randomness as an algorithmic tool. This talk will survey some recent work on the analysis of the well-known Lanczos method for matrix functions and the design of new KSMs for low-rank approximation of matrix functions and approximating partial traces and reduced density matrices. 

Abstract: Fundamental to our understanding of Teichm\"uller space T(S) of a closed oriented genus $g \geq 2$ surface S are two different perspectives: one as connected  component in the  PSL(2,\R) character variety  \chi(\pi_1S, PSL(2,\R)) and one as the moduli space of marked hyperbolic structures on S. The latter can be thought of as a moduli space of (PSL(2,\R), \H^2) -structures. The G-Hitchin component, denoted Hit(S,G), for G a split real simple Lie group, is a connected component in \chi(\pi_1S, G) that is a higher rank generalization of T(S). In this talk, we discuss a new geometric structures (i.e., (G,X)-structures) interpretation of Hit(S, G_2'), where G_2' is the split real form of the exceptional complex simple Lie group G_2.

After discussing the motivation and background, we will present some of the main ideas of the theorem, including a family of almost-complex curves
that serve as bridge between the geometric structures and representations.